-ctg 582 -4tg 492 / ctg 42 +4ctg 498

Лебедев Дмитрий Сергеевич

Учитель математики

Проверено учителем

Для решения выражения ctg 582∘−4tg 492∘ctg 42∘+4ctg 498∘the fraction with numerator ctg 582 raised to the composed with power minus 4 tg 492 raised to the composed with power and denominator ctg 42 raised to the composed with power plus 4 ctg 498 raised to the composed with power end-fraction необходимо привести все тригонометрические функции к одному аргументу, используя формулы приведения и периодичность. 1. Приведение аргументов к острому углу 42∘42 raised to the composed with power Период тангенса и котангенса равен 180∘180 raised to the composed with power. Используем это свойство:

  • ctg 582∘ctg 582 raised to the composed with power:
    582∘=3⋅180∘+42∘582 raised to the composed with power equals 3 center dot 180 raised to the composed with power plus 42 raised to the composed with power.
    Следовательно, ctg 582∘=ctg 42∘ctg 582 raised to the composed with power equals ctg 42 raised to the composed with power. tg 492∘tg 492 raised to the composed with power:
    492∘=2⋅180∘+132∘492 raised to the composed with power equals 2 center dot 180 raised to the composed with power plus 132 raised to the composed with power.
    tg 132∘=tg(180∘−48∘)=−tg 48∘tg 132 raised to the composed with power equals tg open paren 180 raised to the composed with power minus 48 raised to the composed with power close paren equals negative tg 48 raised to the composed with power.
    Используя формулу tg α=ctg(90∘−α)tg alpha equals ctg open paren 90 raised to the composed with power minus alpha close paren, получаем:
    −tg 48∘=−ctg(90∘−48∘)=−ctg 42∘negative tg 48 raised to the composed with power equals negative ctg open paren 90 raised to the composed with power minus 48 raised to the composed with power close paren equals negative ctg 42 raised to the composed with power.
    Итого: tg 492∘=−ctg 42∘tg 492 raised to the composed with power equals negative ctg 42 raised to the composed with power. ctg 498∘ctg 498 raised to the composed with power:
    498∘=2⋅180∘+138∘498 raised to the composed with power equals 2 center dot 180 raised to the composed with power plus 138 raised to the composed with power.
    ctg 138∘=ctg(180∘−42∘)=−ctg 42∘ctg 138 raised to the composed with power equals ctg open paren 180 raised to the composed with power minus 42 raised to the composed with power close paren equals negative ctg 42 raised to the composed with power.
    Итого: ctg 498∘=−ctg 42∘ctg 498 raised to the composed with power equals negative ctg 42 raised to the composed with power.

2. Подстановка значений в выражение Заменим все функции в исходной дроби на их эквиваленты через ctg 42∘ctg 42 raised to the composed with power: ctg 42∘−4(−ctg 42∘)ctg 42∘+4(−ctg 42∘)the fraction with numerator ctg 42 raised to the composed with power minus 4 open paren negative ctg 42 raised to the composed with power close paren and denominator ctg 42 raised to the composed with power plus 4 open paren negative ctg 42 raised to the composed with power close paren end-fraction 3. Упрощение Раскроем скобки в числителе и знаменателе:

  • Числитель: ctg 42∘+4ctg 42∘=5ctg 42∘ctg 42 raised to the composed with power plus 4 ctg 42 raised to the composed with power equals 5 ctg 42 raised to the composed with power Знаменатель: ctg 42∘−4ctg 42∘=-3ctg 42∘ctg 42 raised to the composed with power minus 4 ctg 42 raised to the composed with power equals negative 3 ctg 42 raised to the composed with power

Получаем дробь: 5ctg 42∘-3ctg 42∘the fraction with numerator 5 ctg 42 raised to the composed with power and denominator negative 3 ctg 42 raised to the composed with power end-fraction Сокращаем на ctg 42∘ctg 42 raised to the composed with power (так как ctg 42∘≠0ctg 42 raised to the composed with power is not equal to 0): 5-3=-123≈-1,666...5 over negative 3 end-fraction equals negative 1 and two-thirds is approximately equal to negative 1 comma 666. point point Ответ: -123negative 1 and two-thirds Сообщите, требуется ли вам проверка аналогичных выражений с другими тригонометрическими функциями?

Математика 11 класс

Форма ответа

Ваш аватар

Другие вопросы по теме «Математика»

Вопросы из других предметов